Co-Design of Cryptographic Parameters and Delay-Aware Feedback Gain for Encrypted Control Systems

Abstract: Encrypted control employs homomorphic encryption (HE) to protect both the computation and communication stages, making it a promising approach for secure networked control systems. Most existing results pre-design a controller in the plaintext domain and then implement it over encrypted data. However, this can be problematic because HE induces non-negligible communication and computation delays, which typically increase with the security level, potentially degrading control performance and even destabilizing the closed-loop system. To address this issue, we propose a co-design framework for cryptographic parameters and delay-aware feedback gain. We characterize the encryption-induced delay as a function of the cryptographic parameters and derive a sufficient condition for the existence of a stabilizing delay-aware feedback gain, expressed as a finite set of linear matrix inequalities. This leads to a tractable outer-inner design procedure that searches over cryptographic parameters that satisfy a desired security level and, for each such parameter, seeks a stabilizing feedback gain.

• The paper introduces a co-design framework that jointly optimizes homomorphic encryption parameters and delay-aware feedback gains for encrypted control systems.
• It quantifies encryption-induced delays by decomposing communication and computation components and derives LMI-based stability conditions.
• Empirical results demonstrate that delay-aware gain synthesis maintains system stability under strict security and delay constraints.

Co-Design of Cryptographic Parameters and Delay-Aware Feedback Gain for Encrypted Control Systems

Introduction and Motivation

The paper addresses encrypted control systems employing homomorphic encryption (HE), focusing on the interplay between cryptographic security parameters and control performance in networked control scenarios. Conventional practice designs controllers in the plaintext domain and deploys them within encrypted environments, but this neglects delays sourced from encryption-induced computational and communication overhead. With delays tightly coupled to HE security parameters, ignoring such timing impacts can critically degrade closed-loop performance and even destabilize systems. The research formulates a rigorous co-design framework that quantifies this relationship and enables tractable synthesis of both cryptosystem parameters and delay-aware feedback gains.

Cryptosystem Delay Quantification and Impact

The encrypted control scheme is constructed on a Learning With Errors (LWE)-based cryptosystem, supporting matrix-vector multiplication homomorphically. Security parameters $(N, q, \sigma)$ directly affect computational complexity and ciphertext size, yielding delays in encryption, decryption, homomorphic computation, and network transmission. The paper decomposes the total delay $\tau_\theta$ into communication and computation components, providing tight upper bounds as analytical functions of the cryptographic parameter set $\theta$. Specifically, for $n$-dimensional state and $m$-dimensional input, communication delay scales with $(n+m)(N+1)d/R$, where $d = \lceil \log_2 q \rceil$ and $R$ is transmission rate; computational delays incorporate asymptotic bounds (e.g., $Nd$, $Nd \log N$) per Table 1. The authors rigorously show that security and delay are inherently traded off: higher levels of security (e.g., increasing $\tau_\theta$0, $\tau_\theta$1, $\tau_\theta$2) elevate encryption-induced delays.

Delay-Aware Feedback Gain Synthesis

Control input delays, modeled as constant across sampling instants ($\tau_\theta$3), are explicitly incorporated into the system model by modifying the zero-order hold with delay. System discretization leads to a sampled-data representation with delay-dependent augmented dynamics. The stability analysis focuses on Schur stability of the parameterized augmented system matrix $\tau_\theta$4, where $\tau_\theta$5 is the feedback gain. Due to the continuous parameterization over the interval $\tau_\theta$6, the paper derives a finite tractable criterion via convex polytopic overapproximation and Lyapunov-based Linear Matrix Inequalities (LMIs). The sufficient stability condition requires feasibility of a common symmetric matrix $\tau_\theta$7 for all polytope vertices $\tau_\theta$8.

Co-Design Algorithmic Framework

The co-design problem is posed as a discrete outer-inner loop. The outer loop selects candidate cryptographic parameter sets satisfying a prescribed security requirement (lower bounded via LWE hardness estimator). For each candidate, the delay upper bound is computed and compared to the control sampling interval. Gains $\tau_\theta$9 are synthesized within the inner loop by testing the feasibility of LMIs; only gains meeting stability criteria are accepted. This approach guarantees that controller and cryptosystem jointly satisfy (1) desired security, (2) bounded delay, and (3) stability requirements.

Empirical Evaluation

Simulation results validate the framework by implementing encrypted feedback for a second-order plant using the BGV scheme in Lattigo. Three cryptographic configurations achieving 128-bit security were tested; only one satisfied the delay constraint for a $\theta$0 ms sampling interval (see Table 2). Two feedback gains were synthesized: the delay-unaware $\theta$1 and delay-aware $\theta$2, both via pole placement for delay-free cases. LMI feasibility confirmed $\theta$3 is stabilizing under bounded delay, while $\theta$4 fails as delay increases. Closed-loop state norms across multiple delay values establish the necessity of delay-aware design: $\theta$5 maintains stability, $\theta$6 destabilizes at high delay.

Implications and Outlook

This research establishes a mathematically rigorous co-design paradigm for encrypted control, highlighting the inseparability of security and control performance. Practically, the tractable LMI-based approach is compatible with semidefinite solvers and HE deployment, accommodating real-world cryptosystem constraints. Theoretically, it provides sufficient stability guarantees over a delay-induced convex polytope. The inherent tradeoff between cryptographic strength and responsiveness suggests that secure control architectures must derive cryptosystem parameters in tandem with control design, rather than in isolation. Extensions to time-varying delays, nonlinear feedback, and simultaneous optimization of gain and Lyapunov matrices present fertile ground for advancing encrypted control synthesis.

Conclusion

The paper rigorously addresses joint synthesis of cryptographic system parameters and delay-aware feedback gains in encrypted control systems, quantifying how security demands affect closed-loop stability via induced delays. By deriving a tractable co-design procedure based on explicit delay bounds and LMI stability verification, it establishes a practical and theoretically sound pathway for secure and stable networked control. Future work should generalize to more complex delay models and optimization objectives, further integrating cryptography and control theory.